reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

theorem Th103:
  s in x-succ_of (X, TS) iff ex t st t in X & t, x ==>* s, TS
proof
  thus s in x-succ_of (X, TS) implies ex t st t in X & t, x ==>* s, TS
  proof
    assume s in x-succ_of (X, TS);
    then ex s9 st s9 = s & ex t st t in X & t, x ==>* s9, TS;
    hence thesis;
  end;
  thus thesis;
end;
